Free boundaries surfaces and Saddle towers minimal surfaces in S^2 x R

The aim of this work is to show that for each finite natural number l⩾2 there exists a 1-parameter family of Saddle Tower type minimal surfaces embedded in S^2×R, invariant with respect to a vertical translation. The genus of the quotient surface is 2l−1. The proof is based on analytical techniques: precisely we desingularize of the union of γ_j×R, j∈{1,…,2l}, where γ_j⊂S^2 denotes a half great circle. These vertical cylinders intersect along a vertical straight line and its antipodal line. As byproduct of the construction we produce free boundary surfaces embedded in (S^2)^+×R. Such surfaces are extended by reflection in ∂(S^2)+×R in order to get the minimal surfaces with the desired properties.